Fermionic simulation gates

ABSTRACT

Methods, systems, and apparatus for simulating a physical system. A Hamiltonian describing the physical system is transformed into a qubit Hamiltonian describing a corresponding system of qubits, the qubit Hamiltonian comprising a transformed kinetic energy operator. The evolution of the system of qubits under the qubit Hamiltonian is simulated, including simulating the evolution of the system of qubits under the transformed kinetic energy operator by applying a fermionic swap network to the system of qubits. The simulated evolution of the system of qubits under the qubit Hamiltonian is used to determine properties of the physical system.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a National Stage Application under 35 U.S.C. § 371and claims the benefit of International Application No.PCT/US2017/054714, filed Oct. 2, 2017. The disclosure of the foregoingapplication is hereby incorporated by reference in its entirety.

BACKGROUND

This specification relates to quantum computing.

Quantum simulators are devices designed to provide information andinsights about physical systems or devices. Quantum simulators enablephysical systems that are difficult to study in a laboratory orimpossible to model using a classical processor to be simulated.

SUMMARY

This specification describes methods and systems for implementingfermionic simulation quantum logic gates.

In general, one innovative aspect of the subject matter described inthis specification can be implemented in a method for simulating aphysical system, the method including transforming a Hamiltoniandescribing the physical system into a qubit Hamiltonian describing acorresponding system of qubits, the qubit Hamiltonian comprising atransformed kinetic energy operator; simulating evolution of the systemof qubits under the qubit Hamiltonian, comprising simulating theevolution of the system of qubits under the transformed kinetic energyoperator by applying a fermionic swap network to the system of qubits;and using the simulated evolution of the system of qubits under thequbit Hamiltonian to determine properties of the physical system.

Other implementations of this aspect include corresponding classical orquantum computer systems, apparatus, and computer programs recorded onone or more computer storage devices, each configured to perform theactions of the methods. A system of one or more computers can beconfigured to perform particular operations or actions by virtue ofhaving software, firmware, hardware, or a combination thereof installedon the system that in operation causes or cause the system to performthe actions. One or more computer programs can be configured to performparticular operations or actions by virtue of including instructionsthat, when executed by data processing apparatus, cause the apparatus toperform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the fermionic swap network comprises a quantum circuitcomprising multiple layers of fermionic swap gates.

In some implementations transforming the Hamiltonian describing thephysical system into the corresponding qubit Hamiltonian comprisesapplying the Jordan-Wigner transform.

In some implementations the qubits in the system of qubits are indexedaccording to a canonical ordering, and wherein simulating the evolutionof the system of qubits under the transformed kinetic energy operatorusing the fermionic swap network comprises sequentially applying each ofthe multiple layers of fermionic swap gates to the system of qubits tochange the canonical ordering of the qubits until the canonical orderingis reversed and each index has been adjacent to all others once.

In some implementations the transformed kinetic energy operatorcomprises one or more operators that act on multiple non-adjacent qubitsand wherein sequentially applying each of the multiple layers offermionic swap gates to the system of qubits maps the one or moreoperators that act on multiple non-adjacent qubits to operators that acton two neighboring qubits.

In some implementations the one or more operators that act on multiplenon-adjacent qubits comprise operators of the form X_(p)Z_(p+1)Z_(p+2) .. . Z_(q-1)X_(q) and Y_(p)Z_(p+1)Z_(p+2) . . . Z_(q-1)Y_(q), with X_(p)representing a Pauli-X operator applied to qubit p, Y_(p) representing aPauli-Y operator applied to qubit p, and Z_(p) representing a Pauli-Zoperator applied to qubit p, and wherein the operators that act on twoneighboring qubits comprise operators of the form X_(p)X_(p+1) andY_(p)Y_(p+1).

In some implementations simulating the evolution of the system of qubitsunder the transformed kinetic energy operator using the fermionic swapnetwork further comprises: interleaving gates for simulating theevolution of the system of qubits under the operators that act on twoneighboring qubits between the layers of the fermionic swap network; andapplying the interleaved gates when simulating the evolution of thesystem of qubits.

In some implementations (i) the qubit Hamiltonian further comprises atransformed interaction term comprising operators that act onnon-adjacent qubits in the system of qubits, and (ii) simulating theevolution of the system of qubits under the qubit Hamiltonian furthercomprises simulating the evolution of the system of qubits under thetransformed interaction term using the fermionic swap network,comprising: interleaving gates for simulating the evolution of thesystem of qubits under the operators in the transformed interaction termthat act on neighboring qubits between the layers of the fermionic swapnetwork.

In some implementations the transformed interaction term comprisesoperators of the form (Π−Z_(p)−Z_(p+1)+Z_(p)Z_(P+1)) where Z_(p)represents a Pauli-Z operator applied to qubit p.

In some implementations simulating evolution of the system of qubitsunder the qubit Hamiltonian using the fermionic swap network comprisesapplying a network of fermionic simulation gates to the system ofqubits, the fermionic simulation gates being configured to approximatelysimultaneously (i) simulate the evolution of the system of qubits underthe operator X_(p)X_(p+1)+Y_(p)Y_(p+1) for a time ϕ, (ii) simulateevolution of the system of qubits under the operator Z_(p)Z_(p+1) for atime θ, and (iii) apply a fermionic swap gate to the two modes.

In some implementations N choose 2 fermionic simulation gates aresufficient to implement a Trotter step of evolution, where N representsthe number of qubits in the system of qubits.

In some implementations the fermionic simulation gate is given by

$\left. {{\mathcal{F}\left( {\theta,\phi} \right)} = {\exp\left( {- {i\left\lbrack {{\frac{\pi}{4}\left( {Z_{p} + Z_{p + 1}} \right)} + {\left( {\phi + \frac{\pi}{4}} \right)\left( {{X_{p}X_{p + 1}} + {Y_{p}Y_{p + 1}}} \right)} + {\theta\; Z_{p}Z_{p + 1}} - \frac{\pi}{2}}〛 \right.}} \right\rbrack}} \right).$

In some implementations simulating evolution of the system of qubitsunder the qubit Hamiltonian further comprises applying single qubitrotations to the system of qubits to simulate the single qubit operatorsin the interaction term.

In some implementations (i) the qubit Hamiltonian further comprises atransformed external potential comprising operators that act on singlequbits in the system of qubits, and (ii) simulating evolution of thesystem of qubits under the qubit Hamiltonian further comprisessimulating the evolution of the system of qubits under the transformedexternal potential, comprising applying single qubit rotations to thesystem of qubits.

In some implementations sequentially applying each of the multiplelayers of fermionic swap gates to the system of qubits to change thecanonical ordering of the qubits comprises: indexing the qubits by thecanonical ordering from 1 to N; sequentially applying: fermionic swapgates between odd numbered qubits and even numbered qubits to the right;and fermionic swap gates between even numbered qubits and odd numberedqubits to the right.

In some implementations the physical system comprises a system ofelectrons and wherein the total number of applied layers comprises Napplied layers, with N representing the number of orbitals in the systemof electrons and the number of qubits in the system of qubits.

In some implementations a fermionic swap gate acting on qubit p andqubit q=p+1 is given by

$f_{swap} = {{{JordanWigner}\left\lbrack f_{swap}^{p,q} \right\rbrack} = {\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & {- 1}\end{pmatrix}.}}$

In some implementations the qubit Hamiltonian describes a system ofqubits that interact via linear nearest neighbor couplings.

In some implementations the physical system comprises a system ofelectrons.

In some implementations the system of electrons represent a chemical ormaterial, and wherein using the simulated evolution of the system ofqubits under the qubit Hamiltonian to determine properties of the systemof electrons comprises using the simulated evolution of the system ofqubits under the qubit Hamiltonian to determined properties of thechemical or material.

In some implementations the physical system is described by atwo-dimensional Hubbard model.

The subject matter described in this specification can be implemented inparticular ways so as to realize one or more of the followingadvantages.

A system for simulating electronic structure Hamiltonians usingfermionic simulation gates, as described in this specification, mayachieve increased computational efficiency and require fewercomputational resources compared to other systems for simulatingelectronic structure Hamiltonians.

For example, the systems described in this specification combine thesimulation of the kinetic and interaction terms of an electronicstructure Hamiltonian into a single swap network. Therefore, unlikeother systems for simulating electronic structure Hamiltonians, thesystems described in this specification may avoid using complexprocedures with significant constant overhead, e.g., the fast fermionicFourier transform.

In addition, the systems described in this specification construct andimplement a swap network that uses N layers of fermionic swap gates,with N representing a number of qubits in the system, and requires onlylinear nearest neighbor connectivity between qubits. This providesasymptotical improvements over other systems for simulating electronicstructure Hamiltonians that require that the qubit connectivity graph bea square grid and that use a higher number, e.g.,

${{2N} + {\sqrt{N}\left( {1 + \frac{1}{\sqrt{2}}} \right)} - 2},$of fermionic swap gates.

Furthermore, the swap network used by the system may implementn_(p)n_(q) terms in an electronic structure Hamiltonian withoutrestrictive recursive schemes.

As another example, the systems described in this specification requirefewer quantum logic gates and require only N²/2 entangling two-qubitoperations to simulate systems of electrons—saturating a lower-bound onthe number of two-qubit entangling operations required to implement anexplicit Trotter step of evolution.

In addition, a system for simulating electronic structure Hamiltonians,as described in this specification, simulates electronic Hamiltonians inlinear depth.

A system for simulating electronic structure Hamiltonians, as describedin this specification, may be applied using commercially availablesuperconducting platforms with a gate set of single qubit rotations andCZ gates on a linear array, e.g., simulating jellium with a circuitdepth of 385 gates.

Electronic structure Hamiltonian simulations are relevant to a varietyof industrial applications. For example, simulating electronic structureHamiltonians may form an essential part of developing new batteries,solar cells, catalysts, or superconductors. The techniques forsimulating electronic structure Hamiltonians described in thisspecification may be used to improve and increase the efficiency of suchdevelopment processes.

Fermionic simulation gates, as described in this specification, can beapplied to various other settings in addition to electronic structureHamiltonian simulations. For example, the techniques described in thisspecification can be applied to the Hubbard model to simulate Trottersteps in O(√{square root over (N)}) depth, or can be used to implementvariational algorithms or phase estimation algorithms.

The details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an example device for performing quantum simulationsusing fermionic simulation gates.

FIG. 2 is a diagram of an example circuit for implementing a fermionicsimulation gate.

FIG. 3 is a flow diagram of an example process for simulating a systemof electrons.

FIG. 4 is a flow diagram of an example application of a fermionic swapnetwork.

FIG. 5 is a diagram of an example application of fermionic swap gates ina fermionic swap network.

FIG. 6 is a table showing an example mapping of qubit operators tokinetic energy operator terms at multiple layers of fermionic swapgates.

FIG. 7 is a flowchart of an example process for simulating Trotter stepsof the 2D Hubbard model.

FIG. 8 is an illustration of an example mapping of 2D Hubbard modelsites to a linear qubit chain.

FIG. 9 is an illustration 900 of an example sequence of fermionic swaps.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

This specification describes methods and systems for implementingfermionic simulation gates. One application of fermionic simulationgates is simulating the time evolution of an electronic structureHamiltonian.

The electronic structure Hamiltonian may be expressed in the plane wavedual basis and is given by equation (1) below.

$\begin{matrix}{H = {{\sum\limits_{v,p,q,\sigma}{\frac{k_{v}^{2}{\cos\left( {{k_{v} \cdot r_{q}} - {k_{v} \cdot r_{\rho}}} \right)}}{2N}a_{\rho,\sigma}^{\dagger}a_{q,\sigma}}} - {\underset{j,{v \neq 0}}{\sum\limits_{p,\sigma}}{\frac{4\pi\zeta_{j}{\cos\left( {{k_{v} \cdot R_{j}} - {k_{v} \cdot r_{p}}} \right)}}{\Omega k_{v}^{2}}n_{p,\sigma}}} + {\underset{v \neq 0}{\sum\limits_{{{({p,\sigma})} \neq {({q,\sigma^{\prime}})}},}}{\frac{2\pi{\cos\left( {{k_{v} \cdot r_{\rho}} - {k_{v} \cdot r_{q}}} \right)}}{\Omega k_{v}^{2}}n_{p,\sigma}n_{q,\sigma^{\prime},}}}}} & (1)\end{matrix}$In equation (1), N represents the number of spin-orbitals in the systemof electrons, Ω represents the computational cell volume, d representsthe number of spatial dimensions (e.g., d=3), a_(p,σ) ^(†), a_(p,σ)represent fermionic creation and annihilation operators for spin orbitalϕ_(p,σ),

$p \in \left\lbrack {{- \left( \frac{N}{4} \right)^{\frac{1}{d}}},\left( \frac{N}{4} \right)^{\frac{1}{d}}} \right\rbrack^{d}$represents a vector indexing the spatial orbitals with σ∈{↑,↓} encodingspin, n_(p,q)=a_(p,σ) ^(†),a_(p,σ) represents the number operator forthe corresponding spin orbital,

${r_{p} = {p\left( \frac{2\Omega}{N} \right)}^{\frac{1}{d}}},{k_{v} = {{2\pi\;{v/\Omega^{1/d}}\mspace{14mu}{with}\mspace{14mu} v} \in \left\lbrack {{- \left( \frac{N}{4} \right)^{\frac{1}{d}}},\left( \frac{N}{4} \right)^{\frac{1}{d}}} \right\rbrack^{d}}},$and R_(j) and ξ_(j) represent the position and charge of the j-thnucleus, respectively.

Equation (1) above may be mapped to a system of spins under theJordan-Wigner transform, resulting in a qubit Hamiltonian given byequation (2) below.

$\begin{matrix}{{H = {{\sum\limits_{p,\sigma}{V_{p}Z_{p,\sigma}}} + {\sum\limits_{{({p,\sigma})} \neq {({q,\sigma^{\prime}})}}{V_{pq}Z_{p,\sigma}Z_{q,\sigma^{\prime}}}} + {\sum\limits_{{p \neq q},\sigma}{T_{pq}\left( {{X_{p,\sigma}Z_{{p + 1},\sigma}\mspace{14mu}\ldots\mspace{14mu} Z_{{q - 1},\sigma}X_{q,\sigma}} + {Y_{p,\sigma}Z_{{p + 1},\sigma}\mspace{14mu}\ldots\mspace{14mu} Z_{{q - 1},\sigma}Y_{q,\sigma}}} \right)}}}}\mspace{76mu}{V_{p} = {\sum\limits_{v \neq 0}\left( {\frac{\pi}{\Omega k_{v}^{2}} - \frac{k_{v}^{2}}{4N} + {\sum\limits_{j}\frac{2\pi\zeta_{j}{\cos\left( {{k_{v} \cdot R_{j}} - {k_{v} \cdot r_{p}}} \right)}}{\Omega k_{\nu}^{2}}}} \right)}}\mspace{76mu}{V_{pq} = {\sum\limits_{v \neq 0}\frac{\pi{\cos\left( {{k_{v} \cdot r_{p}} - {k_{v} \cdot r_{q}}} \right)}}{2\Omega k_{\nu}^{2}}}}\mspace{79mu}{T_{pq} = {\sum\limits_{v}\frac{k_{v}^{2}{\cos\left( {{k_{\nu} \cdot r_{p}} - {k_{v} \cdot r_{q}}} \right)}}{4N}}}} & (2)\end{matrix}$In equation (2), X_(p,σ), Y_(p,σ), Z_(p,σ) represent Pauli operatorsthat correspond to spin ½ particles.

Conventional methods for simulating the electronic structure Hamiltonianinclude methods based on Gaussian molecular orbital encoding or theFermionic Fast Fourier Transform (FFFT).

For example, to simulate the evolution of the electronic structureHamiltonian given by equation (1), some conventional techniques simulatesingle Trotter steps of the qubit Hamiltonian given by equation (2) by(i) applying a fermionic fast Fourier transform (FFFT) in order toswitch between the plane dual wave basis, where n_(p)n_(q) are diagonaltwo-qubit operators, and the plane wave basis, where a_(p) ^(†)a_(q) arediagonal single qubit operators, and (ii) applying a linear depth swapnetwork which places all qubits adjacent at least once so that then_(p)n_(q) terms can be simulated. Both steps (i) and (ii) are complexand have significant constant overhead.

This specification describes a new approach to simulate the timeevolution of the electronic structure Hamiltonian. Fermionic swap gatesare used to cycle through different Jordan-Wigner encodings whilesimultaneously performing time evolution under terms of an electronicstructure Hamiltonian describing the system of electrons.

Example Operating Environment

FIG. 1 depicts an example system 100 for performing quantum simulationsusing fermionic simulation gates. The example system 100 is an exampleof a system implemented as classical or quantum computer programs on oneor more classical computers or quantum computing devices in one or morelocations, in which the systems, components, and techniques describedbelow can be implemented.

The system 100 includes quantum hardware 102 in data communication witha classical processor 104. The system 100 may receive as input datarepresenting a system to be simulated, e.g., input data 106. Forexample, the system may receive data representing an electronicstructure Hamiltonian characterizing a system of electrons. An exampleelectronic structure Hamiltonian is given above with reference toequation (1). The system 100 may generate as output data representingresults of simulating the system, e.g., output data 108. For example,the system may output data representing properties of a system ofelectrons.

The system 100 is configured to perform classical computations incombination with quantum computations using the classical processors 104and the quantum hardware 102.

The classical processors 104 are configured to receive the input data106 representing a system to be simulated and to process the receivedinput data 106. Processing the received input data may includetransforming the data to an alternative form that may be simulated usingthe quantum hardware 102.

More specifically, as described above, the input data 106 may includedata representing an electronic structure Hamiltonian. Typically,electronic structure Hamiltonians are expressed in terms of annihilationand creation operators that act on electron states, as described belowwith reference to FIG. 3. The classical processors 104 are configured totransform data representing the electronic structure Hamiltonian to besimulated into data representing a qubit Hamiltonian that characterizesa corresponding system of qubits.

To transform the received input data 106 to the alternative form, theclassical processors 104 may be configured to apply a mapping that mapsannihilation and creation operators to operators that act on qubits. Forexample, the classical processors 104 may apply the Jordan Wignertransformation to data representing an electronic structure Hamiltoniancharacterizing a system of electrons to a qubit Hamiltoniancharacterizing a corresponding system of qubits. The Jordan Wignertransformation is described in more detail below with reference to FIG.3.

The classical processors 104 may provide data representing a qubitHamiltonian characterizing a corresponding system of qubits, e.g.,transformed input data 106, to the quantum hardware 102.

The quantum hardware 102 includes a quantum system 110, one or morecontrol devices 112 and a circuit generator 114. The quantum hardware102 is configured to perform a quantum simulation of the system ofqubits using the transformed input data 106. That is, the quantumhardware 102 may include or be a quantum simulator. To perform a quantumsimulation of the system of qubits using the transformed input data 106,the quantum hardware is configured to generate a quantum circuitrepresenting the qubit Hamiltonian, and apply the generated quantumcircuit to the initialized quantum system 110 using the control devices112. The control devices 112 may then be used to measure the quantumsystem 110 to determine an output of the quantum simulation of thesystem of qubits.

Determined outputs of quantum simulations may be provided to theclassical processors 104. For example, performing a quantum simulationusing the transformed input data may include repeatedly applying thegenerated quantum circuit to the initialized quantum system to obtainmultiple measurement results. The multiple measurement results may beanalyzed by the classical processors 104 to determine statisticalresults of the quantum simulation, e.g., a quantity representing anaverage measurement result. Alternatively or in addition, determinedoutputs of quantum simulations may be post processed by the classicalprocessors 104 to extract relevant information. For example, in somecases observables of interest may comprise a linear combination ofmeasured expectation values. In these examples the classical processors104 may be configured to process the outputs to determine the weightingsof the linear combination (which may be dependent on the encoding of theHamiltonian being simulated).

The quantum system 110 includes a system of qubits 110 a. The system ofqubits 110 a may include qubits that can be efficiently prepared in aninitial state and operated on via application of a quantum circuit. Thetype of physical realizations of qubits included in the quantum systemmay vary. For example, in some implementations the quantum system 110may include superconducting qubits, e.g., superconducting charge qubits,superconducting flux qubits or superconducting phase qubits. In otherimplementations the quantum system 110 may include qubits realized byspins, e.g., electron spins, nuclear spins or atomic spins.

The system of qubits 110 a is arranged as a one-dimensional chain withnearest neighbor interactions. In cases where it is advantageous for thequbits to be arranged as a two or three dimensional array, the two orthree dimensional array may be constructed using the one dimensionalchain (with no intersections of the one dimensional chain) and thetechniques described herein can be equally applied.

The control devices 112 include devices configured to operate on thequantum system 110, e.g., the system of qubits 110 a. For example, thecontrol devices 112 may include hardware for implementing quantum logicgates in a quantum circuit, e.g., microwave control devices. Inaddition, the control devices 112 may include hardware for performingmeasurements on the quantum system 110, e.g., oscillators.

The circuit generator 114 is configured to generate a quantum circuit tobe applied to the quantum system 110. The generated quantum circuitsdepend on the transformed input data.

The qubit Hamiltonian represented by the transformed input data 106 mayinclude three terms—a kinetic energy operator, an interaction term andan external potential. The quantum circuits generated by the circuitgenerator 114 include quantum circuits that simulate the kinetic energyoperator using a fermionic swap network. The fermionic swap networkincludes multiple layers of fermionic swap gates that act on neighboringqubits, where a fermionic swap gate acting on qubits p and q=p+1 isgiven by

$f_{swap} = {{{JordanWigner}\left\lbrack f_{swap}^{p,{q = {p + 1}}} \right\rbrack} = {\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & {- 1}\end{pmatrix}.}}$

Application of the fermionic swap network included in the quantumcircuit generated by the circuit generator 114 maps operators in thequbit Hamiltonian that act on multiple non-adjacent qubits to operatorsthat act on two neighboring qubits. For example, application of thefermionic swap network may map operators in the qubit Hamiltonian of theform X_(p)Z_(p+1)Z_(p+2) . . . Z_(q-1) X_(q) and Y_(p)Z_(p+1)Z_(p+2) . .. Z_(q-1)Y_(q), with X_(p) representing a Pauli-X operator applied toqubit p, Y_(p) representing a Pauli-Y operator applied to qubit p, andZ_(p) representing a Pauli-Z operator applied to qubit p, to operatorsof the form X_(p)X_(p+1) and Y_(p)Y_(p+1). An example fermionic swapnetwork and the action of the fermionic swap gate on a system of qubitsis described in more detail below with reference to FIGS. 3-6.

To simulate the operators that act on two neighboring qubits, e.g.,operators of the form X_(p)X_(p+1) and Y_(p)Y_(p+1), the circuitgenerator 114 interleaves quantum logic gates for simulating theoperators between the layers of the fermionic swap network. The type ofquantum logic gates used to simulate the operators in the kinetic energyoperator that act on two neighboring qubits are illustrated below withreference to FIG. 2. Using a fermionic swap network to simulate akinetic energy operator in a qubit Hamiltonian is described in moredetail below with reference to FIGS. 3-6.

In implementations where the qubit Hamiltonian includes an interactionterm, the quantum circuits generated by the circuit generator 114include quantum circuits that simulate operators in the interaction termthat act on non-adjacent qubits using the fermionic swap network. Forexample, application of the fermionic swap network included in thequantum circuit generated by the circuit generator 114 may map operatorsof the form Z_(p)Z_(q) to operators of the form Z_(p)Z_(p+1). Tosimulate these operators that act on two neighboring qubits, the circuitgenerator 114 interleaves quantum logic gates used to simulate theoperators between the layers of the fermionic swap network. In addition,to simulate operators in the interaction term that act on single qubits,the circuit generator 114 interleaves quantum logic gates used tosimulate the single qubit operators between the layers of the fermionicswap network. The type of quantum logic gates used to simulate theoperators in the interaction term are illustrated below with referenceto FIG. 2. Using a fermionic swap network to simulate an interactionterm in a qubit Hamiltonian is described in more detail below withreference to FIGS. 3-6.

In implementations where the qubit Hamiltonian includes an externalpotential term, the quantum circuits generated by the circuit generator114 include quantum circuits that simulate single qubit operatorsincluded in the external potential by including single qubit rotationsin the quantum circuit. The type of quantum logic gates used to simulatethe operators in the external potential are illustrated below withreference to FIG. 2. Using a fermionic swap network to simulate anexternal potential in a qubit Hamiltonian is described in more detailbelow with reference to FIGS. 3-6.

In some implementations the circuit generator 114 may generate a quantumcircuit that includes one or more fermionic simulation gates 114 a. Afermionic simulation gate is a quantum logic gate that is configured toapproximately simultaneously (i) simulate the evolution of the system ofqubits 110 a under the operator X_(p)X_(p+1)+Y_(p)Y_(p+1) for a time ϕ,(ii) simulate the evolution of the system of qubits 110 a under theoperator Z_(p)Z_(p+1) for a time θ, and (iii) apply a fermionic swapgate to the two modes. A fermionic simulation gate 114 a is given by

${\mathcal{F}\left( {\theta,\phi} \right)} = {\exp{\left. \quad\left( {- {i\left\lbrack {{\frac{\pi}{4}\left( {Z_{p} + Z_{p + 1}} \right)} + {\left( {\phi + \frac{\pi}{4}} \right)\left( {{X_{p}X_{p + 1}} + {Y_{p}Y_{p + 1}}} \right)} + {\theta\; Z_{p}Z_{p + 1}} - \frac{\pi}{2}}〛 \right.}} \right\rbrack \right).}}$An example quantum circuit for implementing a fermionic simulation gate114 a is illustrated below with reference to FIG. 2.

FIG. 2 is a diagram of an example circuit 200 for implementing afermionic simulation gate 218. The example circuit 200 is one examplerealization of a circuit designed to implement a fermionic simulationgate, as described in detail below with reference to FIG. 3. Otherrealizations may also be used to the same effect. The example circuit200 is designed to implement a fermionic simulation gate using atransmon superconducting qubit gate library consisting of arbitrarysingle-qubit rotations and controlled-Z quantum logic gates. However,fermionic simulation gates may be implemented using other qubit gatesets, e.g., corresponding to other qubit realizations.

The fermionic simulation gate 218 is configured to act on two qubitsrepresented by lines p and q. The example circuit 200 for implementingthe fermionic simulation gate 218 includes three entangling operations202, 204 and 206 and seven layers of quantum logic gates. The firstlayer includes two quantum logic gates 220 and 210. Quantum logic gate220 acts on the first qubit p and includes a Hadamard gate H followed bya phase gate S followed by a Hadamard gate H. Quantum logic gate 210acts on the second qubit q and includes a Hadamard gate H.

The second layer includes a first entangling operation 202. The thirdlayer includes two quantum logic gates 222 and 212. Quantum logic gate222 acts on the first qubit p and includes a rotation gate R_(x)(ϕ)followed by a Hadamard gate H. Quantum logic gate 212 acts on the secondqubit q and includes a Hadamard gate H followed by a rotation gate R_(y)(ϕ).

The fourth layer includes a second entangling operation 204. The fifthlayer includes two quantum logic gates 224 and 214. Quantum logic gate224 acts on the first qubit p and includes a Hadamard quantum logic gateH. Quantum logic gate 214 acts on the second qubit q and includes aHadamard gate H followed by the Hermitian conjugate of a phase gate S,followed by a Hadamard gate H, followed by the Hermitian conjugate of aphase gate S, followed by a rotation gate R_(z)(ϕ) followed by aHadamard gate H.

The sixth layer includes a third entangling operation 206. The seventhlayer includes two quantum logic gates 226 and 216. Quantum logic gate226 acts on the first qubit p and includes a phase gate. Quantum logicgate 216 acts on the second qubit q and includes a Hadamard gate Hand aphase gate S.

In cases where the circuit 200 is repeated, the circuit 200 may includesix layers of quantum logic gates since single qubit rotations at theend of one circuit and at the beginning of a subsequent circuit can becombined into one rotation operation.

Using example circuit 200, the total gate depth of implementing oneTrotter step of evolution may be N layers of example circuit 200. Thismay total to a circuit depth of 6N+1, since single qubit gates requiredto simulate single qubit terms in an external potential term andinteraction term of a qubit Hamiltonian can be grouped together as asingle gate and interleaved between each layer.

For example, using the systems and methods described in thisspecification, to simulate a spinless Hamiltonian on a cube with sidelength 4, e.g., N=64 and d=3 dimensions, a Trotter step of evolution maybe only 6×64+1=385 gates deep. In some implementations this may providea number of gates an order of magnitude shallower than the number ofgates required using other methods and systems for simulating electronicstructure Hamiltonians, e.g. those that require 5221 layers. This numberof gates may be within reach of near-term quantum computers and muchsmaller than previous estimates of the gate depth of other quantumsimulation applications—while also being suitable for a linearnearest-neighbor qubit connectivity.

Programming the Hardware

FIG. 3 is a flowchart of an example process 300 for simulating a systemof electrons. For convenience, the process 300 will be described asbeing performed by a system of one or more classical or quantumcomputing devices located in one or more locations. For example, aquantum computation device, e.g., the system 100 of FIG. 1,appropriately programmed in accordance with this specification, canperform the process 300.

The system transforms a Hamiltonian describing the system of electronsinto a qubit Hamiltonian describing a corresponding system of qubits(step 302). Transforming the Hamiltonian describing the system ofelectrons into a qubit Hamiltonian describing a corresponding system ofqubits enables the system to perform a quantum simulation —construct aquantum circuit representing the Hamiltonian describing the system ofqubits, apply the quantum circuit to the system of qubits, and performmeasurements to determine properties about the action of the qubitHamiltonian on the system of qubits, and in turn the action of theelectronic Hamiltonian on the system of electrons.

The Hamiltonian describing the system of electrons may be an electronicstructure Hamiltonian in second quantized form. For example, theHamiltonian may include one or more terms where each term comprisesfermionic annihilation and creation operators a_(p,σ) ^(†) and a_(p,σ)that obey Fermionic anti-commutation relations and describe the actionof removing or adding an electron into spin-orbital ϕ_(p,σ), where prepresents a vector indexing the spatial orbitals of the system ofelectrons and σ encodes electron spin.

The Hamiltonian describing the system of electrons may include a kineticenergy operator, an interaction term and an external potential. Forexample, as described above with reference to equation (1), theHamiltonian describing the system of electrons may be given in the planewave dual basis representation byH _(electron) =T+U+V,where the kinetic energy operator is given by

$\begin{matrix}{{T = {\sum\limits_{p,q,\sigma,v}{\frac{k_{v}^{2}{\cos\left( {{k_{v} \cdot r_{q}} - {k_{v} \cdot r_{p}}} \right)}}{2\; N}a_{p,\sigma}^{\dagger}a_{q,\sigma}}}},} & (3)\end{matrix}$the interaction term is given by

$\begin{matrix}{{U = {\sum\limits_{{{({p,\sigma})} \neq {({q,\sigma^{\prime}})}},{v \neq 0}}{\frac{2\;\pi\;{\cos\left( {{k_{v} \cdot r_{q}} - {k_{v} \cdot r_{q}}} \right)}}{\Omega\; k_{v}^{2}}n_{p,\sigma}n_{q,\sigma^{\prime}}}}},} & (4)\end{matrix}$and the external potential is given by

$\begin{matrix}{V = {- {\sum\limits_{p,\sigma,j,{v \neq 0}}{\frac{4\;\pi\;\xi_{j}{\cos\left( {{k_{v} \cdot R_{j}} - {k_{v} \cdot r_{p}}} \right)}}{\Omega\; k_{v}^{2}}n_{p,\sigma}}}}} & (5)\end{matrix}$where N represents the number of spin-orbitals in the system ofelectrons, Ω represents computational cell volume, d represents thenumber of spatial dimensions (e.g., d=3), n_(p,q)=a_(p,σ) ^(†)a_(p,σ)represents a number operator for a corresponding spin orbital,

${r_{p} = {p\left( \frac{2\;\Omega}{N} \right)}^{\frac{1}{d}}},{k_{v} = {{2\;\pi\;{v/\Omega^{1/d}}\mspace{14mu}{with}\mspace{14mu} v} \in \left\lbrack {{- \left( \frac{N}{4} \right)^{\frac{1}{d}}},\left( \frac{N}{4} \right)^{\frac{1}{d}}} \right\rbrack^{d}}},$and R_(j) and ξ_(j) represent the position and charge of the j-thnucleus, respectively.

To transform the Hamiltonian describing the system of electrons into aqubit Hamiltonian describing a corresponding system of qubits, thesystem may apply a transformation that expresses fermionic operators interms of operators that act on qubits, e.g., quantum logic gates. Forexample, the system may apply the Jordan-Wigner transformation totransform fermionic annihilation and creation operators a_(p,σ) ^(†) anda_(p,σ) to Pauli operators X_(p,σ), Y_(p,σ), Z_(p,σ) that correspond tospin ½ particles. The standard form of the Jordan-Wigner transformationis given below in equation (6).a _(p) ^(†)→½(X _(p) −iY _(p))Z _(p−1) . . . Z ₁ a _(p)→½(X _(p) +iY_(p))Z _(p−1) . . . Z ₁  (6)

Under the Jordan-Wigner transform, the transformed fermionic operatorsdepend on an ordering of the spin orbitals as the action of a fermioniccreation or annihilation operator depends on the occupancies of allorbitals with a lower index. This ordering of the orbitals is referredto as the canonical ordering.

The qubit Hamiltonian describing the system of qubits may include akinetic energy operator, an interaction term, and an external potential.For example, the qubit Hamiltonian may include a transformed kineticenergy operator comprising operators that act on qubits whichcorresponds to the kinetic energy operator comprising fermionicoperators given above in equation (2). Similarly, the qubit Hamiltonianmay include a transformed interaction term and a transformed externalpotential. More specifically, as described above with reference toequation (2), the Hamiltonian describing the system of qubits may be atransformation of an electronic structure Hamiltonian and given byH _(qubit) =T+U+Vwhere the kinetic energy operator is given by

$\begin{matrix}{T = {\sum\limits_{{p \neq q},\sigma}{T_{pq}{\quad{{{\left( {{X_{p,\sigma}Z_{{p + 1},\sigma}\mspace{14mu}\ldots\mspace{14mu} Z_{{q - 1},\sigma}X_{q,\sigma}} + {Y_{p,\sigma}Z_{{p + 1},\sigma}\mspace{14mu}\ldots\mspace{14mu} Z_{{q - 1},\sigma}Y_{q,\sigma}}} \right)\mspace{20mu}{with}\mspace{14mu} T_{pq}} = {\sum\limits_{v}\frac{k_{v}^{2}{\cos\left( {{k_{v} \cdot r_{p}} - {k_{v} \cdot r_{q}}} \right)}}{4\; N}}},}}}}} & (7)\end{matrix}$the interaction term is given by

$\begin{matrix}{{U = {\sum\limits_{{({p,\sigma})} \neq {({q\;\sigma^{\prime}})}}{V_{pq}Z_{p,\sigma}Z_{q,\sigma}}}},{{{with}\mspace{14mu} V_{pq}} = {\sum\limits_{v \neq 0}\frac{\pi\;\cos\;\left( {{k_{v} \cdot r_{p}} - {k_{v} \cdot r_{q}}} \right)}{2\;\Omega\; k_{v}^{2}}}},} & (8)\end{matrix}$and the external potential is given by

$\begin{matrix}{{V = {\sum\limits_{p,\sigma}{V_{p}Z_{p,\sigma}}}}{{{with}\mspace{14mu} V_{p}} = {{\sum\limits_{v \neq 0}\frac{\pi}{\Omega\; k_{v}^{2}}} - \frac{k_{v}^{2}}{4\; N} + {\sum\limits_{j}{\frac{2\;\pi\;\xi_{j}{\cos\left( {{k_{v} \cdot R_{j}} - {k_{v} \cdot r_{p}}} \right)}}{\Omega\; k_{v}^{2}}.}}}}} & (9)\end{matrix}$

In some implementations the system of qubits includes a number of qubitsthat interact via linear nearest neighbor couplings, e.g., aone-dimensional spin chain. In some implementations the linear chain ofqubits may be arranged in the form of a planar lattice or cubic lattice.

The system simulates the evolution of the corresponding system of qubitsunder the qubit Hamiltonian (step 304). Simulating the evolution of thesystem of qubits under the qubit Hamiltonian may include preparing thesystem of qubits in an initial state, constructing a quantum circuitthat represents the qubit Hamiltonian and applying the constructedquantum circuit to the initial state of the system of qubits. In somecases preparing the system of qubits in an initial state may includepreparing the system of qubits in a product state in the plane wavebasis, e.g., in cases where the electronic structure Hamiltonianrepresents a Hamiltonian of the uniform electron gas jellium. Generally,the type of the initial state depends on the Hamiltonian to besimulated.

Simulating the evolution of the corresponding system of qubits under thequbit Hamiltonian includes simulating the system of qubits under theaction of the transformed kinetic energy operator. For example, in caseswhere the Hamiltonian describing the system of electrons is given byequation (1) and the qubit Hamiltonian describing the correspondingsystem of qubits is given by equation (2), simulating the evolution ofthe system of qubits includes simulating the evolution of the system ofqubits under the action of the transformed kinetic energy operator givenby equation (7) above, e.g., under the action of operators that act onmultiple non-adjacent qubits given byX _(p) Z _(p+1) Z _(p+2) . . . Z _(q-1) X _(q) and Y _(p) Z _(p+1) Z_(p+2) . . . Z _(q-1) Y _(q)with X_(p) representing a Pauli-X operator applied to qubit p, Y_(p)representing a Pauli-Y operator applied to qubit p, and Z_(p)representing a Pauli-Z operator applied to qubit p. Simulating theevolution of a system of qubits under such operators is challenging.

Therefore, to simulate the system of qubits under the transformedkinetic energy operator, the system applies a quantum circuit thatcycles through different encodings used to map the electronic structureHamiltonian to the qubit Hamiltonian, e.g., Jordan-Wigner encodings,whilst performing evolution under terms of the qubit Hamiltonian.

The quantum circuit includes a fermionic swap network. The fermionicswap network is a network of quantum logic gates that includes multiplelayers of fermionic swap gates. A fermionic swap gate is a quantum logicgate that is applied to two qubits, e.g., qubits p and q. For example,the fermionic swap gate may be given by f_(swap)=JordanWigner[f_(swap)^(p,q)] where JordanWigner[f_(swap) ^(p,q)] represents a Jordan-Wignertransformation of the fermionic swap operation f_(swap) ^(p,q) operatingon qubits p and q. The fermionic swap operation f_(swap) ^(p,q) isdefined independently of the Jordan-Wigner transformation byf _(swap) ^(p,q)=1+a _(p) ^(†) a _(q) +a _(q) ^(†) a _(p) −a _(p) ^(†) a_(p) −a _(q) ^(†) a _(q).  (10)

The fermionic swap operation exchanges orbitals p and g whilemaintaining proper anti-symmetrization f_(swap) ^(pq,)a_(p)^(†)(f_(swap) ^(p,q))^(†)=a_(q) ^(†). Therefore, the fermionic swapoperation changes the canonical ordering of the qubits, thus alteringwhich fermionic operators are represented by the available two-qubitoperators included in the qubit Hamiltonian.

Since terms of the form n_(p)n_(q) in a Hamiltonian describing a systemof electrons are 2-local, the locality of terms of the form a_(p)^(†)a_(q) is equal to the difference in the canonical ordering index ofp and q. The fermionic swap gates included in the fermionic swap networkmay therefore include fermionic swap gates that are configured to beapplied to neighboring qubits p and q=p+1 under the Jordan-Wignertransform. An example fermionic swap gate that is configured to beapplied to neighboring qubits p and q=p+1 is given by

$\begin{matrix}{f_{swap} = {{{JordanWigner}\left\lbrack f_{swap}^{p,{q = {p + 1}}} \right\rbrack} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & {- 1}\end{pmatrix}}} & (10)\end{matrix}$

The system sequentially applies each of the multiple layers of fermionicswap gates in the fermionic swap network to the system of qubits tochange the canonical ordering of the qubits until the canonical orderingis reversed and each index has been adjacent to all others once. Anexample sequential application of multiple layers of fermionic swapgates is illustrated below with reference to FIG. 5.

Changing the canonical ordering via the fermionic swap network alterswhich fermionic operators are represented by the two-qubit operators inthe qubit Hamiltonian. In particular, the fermionic swap network isconstructed in such a way that each orbital is adjacent to each otherorbital in the canonical ordering at some point in the swap network.Then, at that point in the swap network the fermionic operators whichpreviously mapped to operators that act on multiple non-adjacent qubitsunder the transformation described with reference to step 302, e.g.,X_(p)Z_(p+1)Z_(p+2) . . . Z_(q-1)X_(q) and Y_(p)Z_(p+1)Z_(p+2) . . .Z_(q-1)Y_(q), become operators that act on two neighboring qubits, e.g.,X_(p)X_(p+1) and Y_(p)Y_(p+1).

The evolution of the system of qubits under the transformed kineticenergy operator may therefore be simulated by interleaving gates forsimulating the evolution of the system of qubits under the operatorsthat act on two neighboring qubits, e.g., X_(p) X_(p+1) andY_(p)Y_(p+1), between the layers of the fermionic swap network in thequantum circuit.

Simulating the evolution of the system of qubits under the qubitHamiltonian may further include simulating the system of qubits underthe action of the transformed interaction term. For example, in caseswhere the Hamiltonian describing the system of electrons is given byequation (1) and the qubit Hamiltonian describing the correspondingsystem of qubits is given by equation (2), simulating the evolution ofthe system of qubits includes simulating the evolution of the system ofqubits under the action of the transformed interaction term given byequation (8) above, e.g., under the action of operators that act onmultiple non-adjacent qubits given byZ _(p,σ) Z _(q,σ) ,p≠qwith Z_(p) representing a Pauli-Z operator applied to qubit p.

The evolution of the system of qubits under the transformed interactionterm may also be simulated using the fermionic swap network. Asdescribed above, the fermionic swap network is constructed in such a waythat each orbital is adjacent to each other orbital in the canonicalordering at some point in the swap network. Then, at that point in theswap network the fermionic operators which previously mapped tooperators that act on multiple non-adjacent qubits under thetransformation described with reference to step 302, e.g.,Z_(p,σ)Z_(q,σ), for p≠q, become operators that act on single qubits ortwo neighboring qubits, e.g., (Π−Z_(p)−Z_(p+1)+Z_(p)Z_(p+1)).

The evolution of the system of qubits under the transformed interactionterm may therefore be simulated by interleaving gates for simulating theevolution of the system of qubits under the operators Z_(p)Z_(p+1),between the layers of the fermionic swap network in the quantum circuit.

In some implementations the quantum circuit constructed by the system tosimulate the evolution of the system of qubits under the qubitHamiltonian includes a network of fermionic simulation gates. Afermionic simulation gate is a quantum logic gate that is configured toapproximately simultaneously (i) simulate the evolution of the system ofqubits under two qubit operators X_(p)X_(p+1)+Y_(p)Y_(p+1) for a time ϕ,(ii) simulate evolution of the system of qubits under the two qubitoperator Z_(p)Z_(P+1) for a time ψ, and (iii) apply a fermionic swapgate to the two modes p and q=p+1. The fermionic simulation gate may begiven by

$\left. {{\mathcal{F}\left( {\theta,\phi} \right)} = {\begin{pmatrix}e^{{- i}\;\theta} & 0 & 0 & 0 \\0 & {{- {ie}^{i\;\theta}}\sin\; 2\phi} & {e^{i\;\theta}\cos\; 2\;\phi} & 0 \\0 & {e^{i\;\theta}\cos\; 2\;\phi} & {{- {ie}^{i\;\theta}}\sin\; 2\;\phi} & 0 \\0 & 0 & 0 & {- e^{{- i}\;\theta}}\end{pmatrix} = {\exp\left( {- {i\left\lbrack {{\frac{\pi}{4}\left( {Z_{p} + Z_{p + 1}} \right)} + {\left( {\phi + \frac{\pi}{4}} \right)\left( {{X_{p}X_{p + 1}} + {Y_{p}Y_{p + 1}}} \right)} + {\theta\; Z_{p}Z_{p + 1}} - \frac{\pi}{2}}〛 \right.}} \right\rbrack}}} \right).$

In some implementations

$\quad\begin{pmatrix}N \\2\end{pmatrix}$fermionic simulation gates may be sufficient to implement a Trotter stepof evolution under the qubit Hamiltonian, where N represents the numberof orbitals in the system of electrons and the number of qubits in thesystem of qubits.

When the fermionic simulation gate F(θ, ϕ) acts on adjacent qubitscorresponding to spin orbitals ψ_(p,σ) and ϕ_(q,σ), ϕ=2δ_(σ,σ),T_(pq)and ψ=2V_(pq) where T_(pq) and V_(pq) are defined above with referenceto equations (7) and (8), respectively.

To simulate the single qubit operators included in the interaction term,the system may further include appropriate single qubit rotations in thequantum circuit. For example, the system may include rotations about theZ axis of the Bloch sphere by 2V_(p), where V_(p) is defined above withreference to equation (2), at any point in the quantum circuit.

Simulating the evolution of the system of qubits under the qubitHamiltonian may further include simulating the system of qubits underthe action of the transformed external potential. For example, in caseswhere the Hamiltonian describing the system of electrons is given byequation (1) and the qubit Hamiltonian describing the correspondingsystem of qubits is given by equation (2), simulating the evolution ofthe system of qubits includes simulating the evolution of the system ofqubits under the action of the transformed external potential given byequation (9) above, e.g., under the action of operators that act onsingle qubits given by Z_(p).

In these implementations the system may simulate such single qubitoperators by including appropriate single qubit rotations in the quantumcircuit. For example, the system may include rotations about the Z axisof the Bloch sphere by 2V_(p), where V_(p) is defined above withreference to equation (2), at any point in the quantum circuit.

The system uses the simulated evolution of the system of qubits underthe qubit Hamiltonian to determine properties of the system of electrons(step 306). For example, in cases where the system of electrons to besimulated represent a chemical or a material, using the simulatedevolution of the system of qubits under the qubit Hamiltonian todetermine properties of the system of electrons may include using thesimulated evolution of the system of qubits under the qubit Hamiltonianto determine properties of the chemical or material, e.g., theconductance of a metal, chemical dynamics, elastic moduli, energy,dipole moment, or optical properties.

FIG. 4 is a flow diagram of an example application of a fermionic swapnetwork. For convenience, the process 400 will be described as beingperformed by a system of one or more classical or quantum computingdevices located in one or more locations. For example, a quantumcomputation system, e.g., the system 100 of FIG. 1, appropriatelyprogrammed in accordance with this specification, can perform theprocess 400.

The system indexes the qubits using the Jordan-Wigner canonical orderingfrom 1 to N (step 402). For example, in cases where the qubits arearranged in a linear array, i.e., a line, a first qubit in the lineararray may correspond to index i=1 and a last qubit in the linear arraymay correspond to index i=N.

The system sequentially applies fermionic swap gates between oddnumbered qubits and even numbered qubits to the right (step 404) andfermionic swap gates between even numbered qubits and odd numberedqubits to the right (step 406).

More specifically, the system alternately applies (1) fermionic swapgates between odd qubits 2i+1 and even qubits to their right 2i+2 fori∈[0, (N−2)/2], and (2) even qubits 2i+2 and odd qubits to their right2i+3 for i∈[0, (N−2)/2]. At each application of stage (1), the lastqubit in the indexed line of qubits is untouched for odd N, e.g., as isthe case for a spinless Hamiltonian, because there is no even numberedqubit to its right. At each application of stage (2), the first qubit inthe indexed line of qubits is untouched for N even or odd because thereis no qubit on its left. The last qubit is also untouched for even N.

The system sequentially applies the fermionic swap gates until thecanonical ordering of the qubits is reversed. Example canonical orderingchanges through five layers of fermionic swap gates are illustratedbelow with reference to FIG. 5.

FIG. 5 is a diagram 500 of an example application of fermionic swapgates in a fermionic swap network. For convenience, the diagram 500shows an example application of fermionic swap gates in a fermionic swapnetwork with N=5 layers (that are each configured to be applied to N=5qubits). However, the number of layers (and qubits) included in thefermionic swap network is dependent on the number of spin orbitals inthe system of electrons that is to be simulated and may therefore vary.

The diagram 500 shows how the canonical ordering of five qubits may beupdated through the application of a fermionic swap network thatincludes five layers 521-525. Each circle 1-5 represents a qubit in alinear array. The qubits do not move—the ψ_(p) inside each circlerepresents which spin-orbital occupancy is encoded by the respectivequbit during a particular fermionic swap network layer. For example, atlayer 521 qubit 1 encodes spin-orbital occupancy ψ₁, at layers 522 and523 qubit 1 encodes spin-orbital occupancy ψ₂, at layers 525 qubit 1encodes spin-orbital occupancy ψ₄ and at the final layer 526 qubit 1encoders spin-orbital occupancy ψ₅.

The lines between qubits, e.g., line 502 between qubits 1 and 2(hereafter denoted as (1, 2)) and line 504 between qubits (3, 4),represent fermionic swap gates, as described above in equation (10),which change the canonical ordering of the linear array of qubits sothat different spin-orbital occupancies are represented by differentqubits in a subsequent layer.

In the first layer 521 of the fermionic swap network, theone-dimensional chain of qubits 1-5 are indexed according to an examplecanonical ordering ψ₁, ψ₂, ψ₃, ψ₄, ψ₅. The system then applies fermionicswap gates 502 and 504 between qubits (1, 2) and (3, 4) respectively.This application of fermionic swap gates results in an updated canonicalordering ψ₂, ψ₁, ψ₄, ψ₃, ψ₅.

In the second layer 522 of the fermionic swap network, the systemapplies fermionic swap gates between qubits (1, 4) and (3, 5). Thisapplication of fermionic swap gates results in an updated canonicalordering ψ₂, ψ₄, ψ₁, ψ₅, ψ₃.

In the third layer 523 of the fermionic swap network, the system appliesfermionic swap gates between qubits (2, 4) and (1, 5). This applicationof fermionic swap gates results in an updated canonical ordering ψ₄, ψ₂,ψ₅, ψ₁, ψ₃.

In the fourth layer 524 of the fermionic swap network, the systemapplies fermionic swap gates between qubits (2, 5) and (1, 3). Thisapplication of fermionic swap gates results in an updated canonicalordering ψ₄, ψ₅, ψ₂, ψ₃, ψ₁.

In the fifth layer of the fermionic swap network, the system appliesfermionic swap gates between qubits (4, 5) and (2, 3). This applicationof fermionic swap gates results in an updated canonical ordering ψ₅, ψ₄,ψ₃, ψ₂, ψ₁ that is a reverse of the canonical ordering ψ₁, ψ₂, ψ₃, ψ₄,ψ₅, as shown in the final row 526.

The total number of sequential applications of fermionic swap gates,i.e., the total number of layers of fermionic swap gates applied to thesystem of qubits is N=5, where N represents the number of qubits in thesystem of qubits (and the number of spin-orbitals in the system ofelectrons to which the system of qubits corresponds to).

FIG. 6 is a table 600 showing an example mapping of qubit operators tokinetic energy operator terms at the multiple layers of fermionic swapgates described above with reference to FIG. 5. For example, rows604-612 of table 600 correspond to layers 521-525, respectively, ofdiagram 500 with reference to FIG. 5.

The entries of the first column 602 of table 600 indicates which pairsof qubits are involved in respective fermionic swaps gates. For example,in row 604 of table 600, column 614 indicates that in the first layer521 of FIG. 5, fermionic swap gates are applied to qubits 1 and 2encoding spin-orbital occupancy ψ₁ and ψ₂, respectively and to qubits 3and 4 encoding spin-orbital occupancy ψ₃ and ψ₄, respectively. Asanother example, in row 606 of table 600, column 614 indicates that inthe second layer 522 of FIG. 5, fermionic swap gates are applied toqubits 2 and 3 encoding spin-orbital occupancy ψ₁ and ψ₄, respectivelyand to qubits 4 and 5 encoding spin-orbital occupancy ψ₃ and ψ₅,respectively.

The entries of columns 614, 616, 618 and 620 indicate which kineticenergy fermionic operators a_(p) ^(†), a_(p) the qubit operators X_(p),Y_(p) correspond to at each layer. For example, the entry correspondingto row 604 and column 614 indicates that the qubit operators ½(X₁X₂+Y₁Y₂) correspond to kinetic energy fermionic operators a₁^(†)a₂+a₂ ^(†)a₁ at layer 521 in diagram 500 of FIG. 5. As anotherexample, the entry corresponding to row 606 and column 616 indicatesthat the qubit operators ½ (X₂X₃+Y₂Y₃) correspond to kinetic energyfermionic operators a₁ ^(†)a₄+a₄ ^(†)a₁ at layer 522 in diagram 500 ofFIG. 5.

Table 600 shows that, as described above with reference to step 304 ofFIG. 3,

$\begin{pmatrix}N \\2\end{pmatrix} = {\begin{pmatrix}5 \\2\end{pmatrix} = {10}}$possible kinetic energy interactions appear in the table (since diagonallines through entries of the table indicate that the pair of qubits isnot acted on in that layer) and are this applied during application ofthe fermionic swap network. Since the fermionic simulation gatedescribed above with reference to FIG. 3 also applies the potentialenergy interactions during application of the fermionic swap network,the five layers 521-525 represented by rows 604-612 may implement anentire Trotter step of evolution.

In addition to the simulation of electronic structures, the methods andsystems described in this specification may be applied to various othersettings and tasks. For example, quantum circuits that include fermionicsimulation gates can be applied to the 2D Hubbard model to simulateTrotter steps in O(√{square root over (N)}) depth on a linear array ofqubits, as described in detail below with reference to FIGS. 7-9.

As another example, quantum circuits that include fermionic simulationgates can be used to form a quantum variational algorithm based onTrotter steps. For example, the fermionic simulation gate schemesdescribed herein may be used to apply Trotter steps to a quantum stateof a quantum system, resulting in a quantum state that is a variationalapproximation to a quantum state of interest. The energy (or anotherobservable) of the quantum system can then be measured and used toadjust system parameters to minimize the energy of the quantum system.In some cases such a variational algorithm may be implemented on a nearterm quantum computing device.

As another example, the techniques described in this specification maybe used to perform an exact quantum phase estimation algorithm incombination with error correction algorithms, e.g., error correctedquantum simulations. More specifically, the quantum phase estimationalgorithm determines a ground state of a quantum system exactly byrepeatedly applying a unitary operator to the quantum system. Thefermionic simulation gate schemes described herein may be used incombination with Trotterization to approximate the unitary operator.

FIG. 7 is a flowchart of an example process 700 for simulating Trottersteps of the 2D Hubbard model. For convenience, the example process 700is described as simulating Trotter steps of the 2D Hubbard model withspins, given by

$H = {{{- t}{\sum\limits_{{\langle{pq}\rangle},\sigma}\left( {{a_{p,\sigma}^{\dagger}a_{q,\sigma}} + {a_{q,\sigma}^{\dagger}a_{p,\sigma}}} \right)}} + {U{\sum\limits_{p}{n_{p, \uparrow}{n_{p, \downarrow}.}}}}}$However, the described process may also be applied to a spinless 2DHubbard model, e.g., a spinless Hubbard model with an odd number ofsites in each row. In addition, the described process may also beapplied to 3D models (resulting in O(N^(2/3) gate depth).

The process 700 will be described as being performed by a system of oneor more classical or quantum computing devices located in one or morelocations. For example, a quantum computation device, e.g., the system100 of FIG. 1, appropriately programmed in accordance with thisspecification, can perform the process 700.

The system uses the Jordan-Wigner transformation to map the 2D Hubbardmodel Hamiltonian with spins to a qubit Hamiltonian characterizing acorresponding system of qubits (step 702). Mapping the 2D Hubbard modelHamiltonian to a qubit Hamiltonian includes choosing a particularordering of the orbitals for the Jordan Wigner transformation, asillustrated below with reference to FIG. 8.

The system performs a series of fermionic swaps on the system of qubits(step 704). The series of fermionic swaps are illustrated below withreference to FIG. 9.

FIG. 8 is an illustration 800 of an example mapping of 2D Hubbard modelsites to a linear qubit chain. Each circle in illustration 800represents a spin-orbital. Circles drawn with a dashed line, e.g.,circle 802, represents a spin-up orbital. Circles drawn with a solidline, e.g., circle 804, represents a spin-down orbital. Arrows betweencircles represent the canonical ordering that should be used in theJordan Wigner transformation. The illustration 800 shows that thecanonical ordering includes alternating whether the up or down orbitalcomes first across the rows and alternating whether to order inascending or descending across columns.

FIG. 9 is an illustration 900 of two example layers 900 a and 900 b offermionic swaps. In both layer 900 a and 900 b, each circle represents aspin-orbital. Circles drawn with a dashed line, e.g., circle 902,represents a spin-up orbital. Circles drawn with a solid line, e.g.,circle 904, represents a spin-down orbital. Arrows between circles,e.g., arrow 906, represent applications of fermionic swap gates.

By repeatedly applying the layers 900 a and 900 b in a particular way itis possible to bring spin orbitals from adjacent rows next to each otherin the canonical ordering so that the hopping term in the Hubbard modelHamiltonian may be applied locally.

Repeatedly applying the layers 900 a (U_(L)) and 900 b (U_(R)) includesfirst applying U_(L). This enables application of the remaininghorizontal hopping term that could not be previously reached. Then,U_(R) U_(L) is repeatedly applied. After each application of U_(R) U_(L)new vertical hopping terms become available until U_(R) U_(L) has beenapplied a total of √{square root over (N/8)}−1 times, with Nrepresenting the number of qubits in the system of qubits. At thispoint, the series of swaps needs to be reversed until the orbitals areback to their original locations in the canonical ordering. At thispoint, applying U_(R) U_(L) causes the qubits to circulate in the otherdirection. The may be repeated for a total of √{square root over(N/8)}−1 times to ensure that all neighboring orbitals are adjacent atleast once. The total number of layers of fermionic swaps required forthe whole procedure is √{square root over (9N/2)}.

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible. The term “dataprocessing apparatus” refers to digital and/or quantum data processinghardware and encompasses all kinds of apparatus, devices, and machinesfor processing digital and/or quantum data, including by way of examplea programmable digital processor, a programmable quantum processor, adigital computer, a quantum computer, multiple digital and quantumprocessors or computers, and combinations thereof. The apparatus canalso be, or further include, special purpose logic circuitry, e.g., anFPGA (field programmable gate array), an ASIC (application-specificintegrated circuit), or a quantum simulator, i.e., a quantum dataprocessing apparatus that is designed to simulate or produce informationabout a specific quantum system. In particular, a quantum simulator is aspecial purpose quantum computer that does not have the capability toperform universal quantum computation. The apparatus can optionallyinclude, in addition to hardware, code that creates an executionenvironment for digital and/or quantum computer programs, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer mayreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that may include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

The invention claimed is:
 1. A method for simulating a physical system,the method comprising: transforming a Hamiltonian describing thephysical system into a qubit Hamiltonian describing a correspondingsystem of qubits, the qubit Hamiltonian comprising a transformed kineticenergy operator; simulating evolution of the system of qubits under thequbit Hamiltonian, comprising simulating the evolution of the system ofqubits under the transformed kinetic energy operator by applying afermionic swap network to the system of qubits; and using the simulatedevolution of the system of qubits under the qubit Hamiltonian todetermine properties of the physical system.
 2. The method of claim 1,wherein the fermionic swap network comprises a quantum circuitcomprising multiple layers of fermionic swap gates.
 3. The method ofclaim 2, wherein transforming the Hamiltonian describing the physicalsystem into the corresponding qubit Hamiltonian comprises applying theJordan-Wigner transform.
 4. The method of claim 3, wherein the qubits inthe system of qubits are indexed according to a canonical ordering, andwherein simulating the evolution of the system of qubits under thetransformed kinetic energy operator using the fermionic swap networkcomprises sequentially applying each of the multiple layers of fermionicswap gates to the system of qubits to change the canonical ordering ofthe qubits until the canonical ordering is reversed and each index hasbeen adjacent to all others once.
 5. The method of claim 4, wherein thetransformed kinetic energy operator comprises one or more operators thatact on multiple non-adjacent qubits and wherein sequentially applyingeach of the multiple layers of fermionic swap gates to the system ofqubits maps the one or more operators that act on multiple non-adjacentqubits to operators that act on two neighboring qubits.
 6. The method ofclaim 5, wherein the one or more operators that act on multiplenon-adjacent qubits comprise operators of the form X_(p)Z_(p+1)Z_(p+2) .. . Z_(q-1)X_(q) and Y_(p)Z_(p+1)Z_(p+2) . . . Z_(q-1)Y_(q), with X_(p)representing a Pauli-X operator applied to qubit p, Y_(p) representing aPauli-Y operator applied to qubit p, and Z_(p) representing a Pauli-Zoperator applied to qubit p, and wherein the operators that act on twoneighboring qubits comprise operators of the form X_(p)X_(p+1) andY_(p)Y_(p+1).
 7. The method of claim 5, wherein simulating the evolutionof the system of qubits under the transformed kinetic energy operatorusing the fermionic swap network further comprises: interleaving gatesfor simulating the evolution of the system of qubits under the operatorsthat act on two neighboring qubits between the layers of the fermionicswap network; and applying the interleaved gates when simulating theevolution of the system of qubits.
 8. The method of claim 7, wherein (i)the qubit Hamiltonian further comprises a transformed interaction termcomprising operators that act on non-adjacent qubits in the system ofqubits, and (ii) simulating the evolution of the system of qubits underthe qubit Hamiltonian further comprises simulating the evolution of thesystem of qubits under the transformed interaction term using thefermionic swap network, comprising: interleaving gates for simulatingthe evolution of the system of qubits under the operators in thetransformed interaction term that act on neighboring qubits between thelayers of the fermionic swap network.
 9. The method of claim 8, whereinthe transformed interaction term comprises operators of the form(Π−Z_(p)−Z_(p+1)+Z_(p)Z_(P+1)) where Z_(p) represents a Pauli-Z operatorapplied to qubit p.
 10. The method of claim 9, wherein simulatingevolution of the system of qubits under the qubit Hamiltonian using thefermionic swap network comprises applying a network of fermionicsimulation gates to the system of qubits, the fermionic simulation gatesbeing configured to approximately simultaneously (i) simulate theevolution of the system of qubits under the operatorX_(p)X_(p+1)+Y_(p)Y_(p+1) for a time ϕ, (ii) simulate evolution of thesystem of qubits under the operator Z_(p)Z_(P+1) for a time θ, and (iii)apply a fermionic swap gate to the two modes p and p+1.
 11. The methodof claim 10, wherein N choose 2 fermionic simulation gates aresufficient to implement a Trotter step of evolution, where N representsthe number of qubits in the system of qubits.
 12. The method of claim10, wherein the fermionic simulation gate is given by${\mathcal{F}\left( {\theta,\phi} \right)} = {\exp{\left. \quad\left( {- {i\left\lbrack {{\frac{\pi}{4}\left( {Z_{p} + Z_{p + 1}} \right)} + {\left( {\phi + \frac{\pi}{4}} \right)\left( {{X_{p}X_{p + 1}} + {Y_{p}Y_{p + 1}}} \right)} + {\theta\; Z_{p}Z_{p + 1}} - \frac{\pi}{2}}〛 \right.}} \right\rbrack \right).}}$13. The method of claim 9, wherein simulating evolution of the system ofqubits under the qubit Hamiltonian further comprises applying singlequbit rotations to the system of qubits to simulate the single qubitoperators in the interaction term.
 14. The method of claim 1, wherein(i) the qubit Hamiltonian further comprises a transformed externalpotential comprising operators that act on single qubits in the systemof qubits, and (ii) simulating evolution of the system of qubits underthe qubit Hamiltonian further comprises simulating the evolution of thesystem of qubits under the transformed external potential, comprisingapplying single qubit rotations to the system of qubits.
 15. The methodof claim 4, wherein sequentially applying each of the multiple layers offermionic swap gates to the system of qubits to change the canonicalordering of the qubits comprises: indexing the qubits by the canonicalordering from 1 to N; sequentially applying: fermionic swap gatesbetween odd numbered qubits and even numbered qubits to the right; andfermionic swap gates between even numbered qubits and odd numberedqubits to the right.
 16. The method of claim 15, wherein the physicalsystem comprises a system of electrons and wherein the total number ofapplied layers comprises N applied layers, with N representing thenumber of orbitals in the system of electrons and the number of qubitsin the system of qubits.
 17. The method of claim 2, wherein a fermionicswap gate acting on qubit p and qubit q=p+1 is given by$f_{swap} = {{{JordanWigner}\left\lbrack f_{swap}^{p,q} \right\rbrack} = {\begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & {- 1}\end{pmatrix}.}}$
 18. The method of claim 1, wherein the qubitHamiltonian describes a system of qubits that interact via linearnearest neighbor couplings.
 19. The method of claim 1, wherein thephysical system comprises a system of electrons.
 20. The method of claim19, wherein the system of electrons represent a chemical or material,and wherein using the simulated evolution of the system of qubits underthe qubit Hamiltonian to determine properties of the system of electronscomprises using the simulated evolution of the system of qubits underthe qubit Hamiltonian to determined properties of the chemical ormaterial.
 21. The method of claim 1, wherein the physical system isdescribed by a two-dimensional Hubbard model.
 22. An apparatuscomprising: quantum hardware, comprising: a plurality of qubits, aplurality of single qubit gates, a plurality of two qubit gates; one ormore classical processors; wherein the apparatus is configured toperform operations comprising: transforming, by the classicalprocessors, a Hamiltonian describing a physical system into a qubitHamiltonian describing a corresponding system of qubits, the qubitHamiltonian comprising a transformed kinetic energy operator;simulating, by the quantum hardware, evolution of the system of qubitsunder the qubit Hamiltonian, comprising simulating the evolution of thesystem of qubits under the transformed kinetic energy operator byapplying a fermionic swap network to the system of qubits, wherein thefermionic swap network is constructed from the quantum hardware; andusing the simulated evolution of the system of qubits under the qubitHamiltonian to determine properties of the physical system.